Rangers became the first Scottish club to win a European tie in Italy with a sumptuous display to sink Livorno.
Nacho Novo scored Rangers' third goal against Livorno
Charlie Adam fired a stunning volley on 27 minutes and Kris Boyd made it two from the penalty spot on 30 minutes.
Livorno's Cristiano Lucarelli scored with a controversial penalty on 32 minutes after Julien Rodriguez was judged to have fouled Luca Vigiani.
Soon afterwards Nacho Novo drilled home Boyd's headed pass, before Lucarelli scrambled home his second late on.
Rangers manager Paul Le Guen opted to ditch his favoured formation this season and played Novo and Boyd up front, with the waspish Thomas Buffel also in an advanced role.
Both teams struggled early on in the treacherous conditions as the torrential rain made fluent passing difficult.
Rangers goalkeeper Lionel Letizi looked unconvincing in the opening 10 minutes as he flapped at a cross ball and almost handled outside the box.
Boyd headed wide after 14 minutes following clever work by Buffel on the right-wing.
A minute later Novo shot wide after getting into a dangerous position 25 yards from goal.
Adam scored the opening goal on 27 minutes after he volleyed home from 20 yards after taking a headed pass from Boyd.
Penalty goalscorer Kris Boyd is congratulated by his team mates
Rangers capitalised on their early lead when Novo was fouled in the box by Samuel Kuffour after a powerful run.
Boyd took the penalty and sent the keeper the wrong way to give the Ibrox side a two-goal lead.
Livorno were handed a lifeline on 32 minutes in controversial circumstances when they were awarded a penalty.
Lucarelli scored from the spot after Rodriguez was judged to have fouled Vigiani.
Novo restored Rangers' two goal advantage after he drilled home another Boyd headed pass from 20 yards.
Barry Ferguson was booked on 42 minutes for kicking the ball away after conceding a foul.
Jose Vidigal joined the Rangers captain in the book for a reckless challenge on Smith on the stroke of half-time.
In the second half Livorno battled to reduce the deficit but despite enjoying long spells of possession they could not break down the Rangers rearguard until late on.
Letizi made a terrific save on the hour from Danilevicius' diving header.
The Ibrox side could have made it four when Ferguson opted to shoot rather than play in Novo after the Spaniard had made a great run.
Filip Sebo replaced Boyd with 10 minutes to go as Rangers were content to play out time.
Lucarelli scrambled home Livorno's second in injury time but Rangers held on for a deserved victory.
Livorno: Amelia, Galante, Vidigal, Vigiani (Bakayoko 78), Kuffour, Rezaei (Morrone 57), Pasquale, Passoni, Cesar Prates (Balleri 46), Danilevicius, Lucarelli.
Subs Not Used: Manitta, Filippini, Pfertzel, Grandoni.
Booked: Vidigal, Kuffour.
Goals: Lucarelli 34 pen, 90.
Rangers: Letizi, Buffel (Rae 89), Ferguson, Hemdani, Novo (Prso 87), Kris Boyd (Sebo 82), Svensson, Hutton, Adam, Smith, Rodriguez.
Subs Not Used: McGregor, Ponroy, Stanger, N'Diaye.
Booked: Ferguson, Hutton.
Goals: Adam 27, Kris Boyd 30 pen, Novo 35.
Ref: Julian Rodriguez Santiago (Spain).